A circuit consists of a light bulb, a capacitor, and an

You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you can run through this bulb?

A sinusoidal current has an rms value (a) What is the current amplitude? (b) The current is passed through a full-wave rectifier circuit. What is the rectified average current? (c) Which is larger: or Explain, using graphs of and of the rectified current.

The voltage across the terminals of an ac power supply varies with time according to Eq. (31.1). The voltage amplitude is V = 45.0 V. What are (a) the root-mean-square potential difference \(V_{\mathrm{rms}}\) and (b) the average potential difference \(V_{\mathrm{av}}\) between the two terminals of the power supply?

A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. (a) What is the phase angle for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What is the capacitance C of the capacitor if the current amplitude is 5.30 A?

An inductor with is connected across an ac source that has voltage amplitude 45.0 V. (a) What is the phase angle for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What value for the frequency of the source results in a current amplitude of 3.90 A?

A capacitance and an inductance are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If and what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Kitchen Capacitance. The wiring for a refrigerator contains a starter capacitor. A voltage of amplitude 170 V and frequency 60.0 Hz applied across the capacitor is to produce a current amplitude of 0.850 A through the capacitor. What capacitance is required?

(a) Compute the reactance of a 0.450-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450-H inductor equal to that of a capacitor?

(a) What is the reactance of a 3.00-H inductor at a frequency of 80.0 Hz? (b) What is the inductance of an inductor whose reactance is at 80.0 Hz? (c) What is the reactance of a capacitor at a frequency of 80.0 Hz? (d) What is the capacitance of a capacitor whose reactance is at 80.0 Hz?

A Radio Inductor. You want the current amplitude through a 0.450-mH inductor (part of the circuitry for a radio receiver) to be 2.60 mA when a sinusoidal voltage with amplitude 12.0 V is applied across the inductor. What frequency is required?

A 0.180-H inductor is connected in series with a \(90.0-\Omega\) resistor and an ac source. The voltage across the inductor is \(v_{L}=-(12.0 \mathrm{~V}) \sin [(480 \mathrm{rad} / \mathrm{s}) t]\) . (a) Derive an expression for the voltage across the resistor. (b) What is \(v_{R}\) at t=2.00ms ?

A \(150-\Omega\) resistor is connected in series with a 0.250-H capacitor and an ac source. The voltage across the capacitor is \(V_{R} = (3.80 V) cos[ (720 rad/s)t]\). (a) Determine the capacitive reactance of the capacitor. (b) Derive an expression for the voltage \(v_{L}\) across the resistor. Text Transcription: 150-omega v_R = (3.80 V) cos [(720 rad/s) t] v_L

A resistor is connected in series with a 0.250-H inductor and an ac source. The voltage across the resistor is (a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage across the inductor.

You have a resistor, a 0.400-H inductor, and a capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

The resistor, inductor, capacitor, and voltage source described in Exercise 31.14 are connected to form an L-R-C series circuit. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What is the phase angle of the source voltage with respect to the current? Does the source voltage lag or lead the current? (d) What are the voltage amplitudes across the resistor, inductor, and capacitor? (e) Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source.

A resistor, a 0.900-H inductor, and a capacitor are connected in series across a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad s. (a) What are , , , and at ? Compare to at this instant. (b) What are , , and ? Compare V to . Explain why these two quantities are not equal.

In an L-R-C series circuit, the rms voltage across the resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor it is 50.0 V. What is the rms voltage of the source?

A resistor with \(R = 300 \Omega\) and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 216 W. (a) What is the impedance Z of the circuit? (b) What is the amplitude of the voltage across the inductor? (c) What is the power factor? Text Transcription: R = 300 omega

The power of a certain CD player operating at 120 V rms is 20.0 W. Assuming that the CD player behaves like a pure resistor, find (a) the maximum instantaneous power; (b) the rms current; (c) the resistance of this player.

In an L-R-C series circuit, the components have the following values: and The generator has an rms voltage of 120 V and a frequency of 1.25 kHz. Determine (a) the power supplied by the generator and (b) the power dissipated in the resistor

(a) Show that for an series circuit the power factor is equal to . (b) An series circuit has phase angle . The voltage amplitude of the source is 90.0 V. What is the voltage amplitude across the resistor?

(a) Use the results of part (a) of Exercise 31.21 to show that the average power delivered by the source in an series circuit is given by . (b) An series series circuit has , and the amplitude of the voltage across the resistor is 36.0 V. What is the average power delivered by the source?

An L-R-C series circuit with L = 0.120 H, \(R=240\ \Omega\) and \(C=7.30\ \mu\mathrm{F}\) carries an rms current of 0.450 A with a frequency of 400 Hz. (a) What are the phase angle and power factor for this circuit? (b) What is the impedance of the circuit? (c) What is the rms voltage of the source? (d) What average power is delivered by the source? (e) What is the average rate at which electrical energy is converted to thermal energy in the resistor? (f) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? (g) In the inductor?

An L-R-C series circuit is connected to a 120-Hz ac source that has The circuit has a resistance of and an impedance at this frequency of What average power is delivered to the circuit by the source?

A series ac circuit contains a resistor, a 15-mH inductor, a capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

In an L-R-C series circuit the source is operated at its resonant angular frequency. At this frequency, the reactance \(X_{C}\) of the capacitor is \(200\ \Omega\) and the voltage amplitude across the capacitor is 600 V. The circuit has \(R=300\ \Omega\) . What is the voltage amplitude of the source?

Analyzing an L-R-C Circuit. You have a resistor, a 0.400-H inductor, a capacitor, and a variablefrequency ac source with an amplitude of 3.00 V. You connect all four elements together to form a series circuit. (a) At what frequency will the current in the circuit be greatest? What will be the current amplitude at this frequency? (b) What will be the current amplitude at an angular frequency of At this frequency, will the source voltage lead or lag the current?

An L-R-C series circuit is constructed using a resistor, a capacitor, and an 8.00-mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 V. (a) At what angular frequency will the impedance be smallest, and what is the impedance at this frequency? (b) At the angular frequency in part (a), what is the maximum current through the inductor? (c) At the angular frequency in part (a), find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value. (d) In part (c), how are the potential differences across the resistor, inductor, and capacitor related to the potential difference across the ac source?

In an L-R-C series circuit, \(R=300\ \Omega,\ L=0.400 \mathrm{\ H}\), and \(C=6.00 \times 10^{-8} \mathrm{\ F}\). When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 A. (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

An L-R-C series circuit consists of a source with voltage amplitude 120 V and angular frequency a resistor with , an inductor with and a capacitor with capacitance (a) For what value of will the current amplitude in the circuit be a maximum? (b) When has the value calculated in part (a), what is the amplitude of the voltage across the inductor?

In an L-R-C series circuit, and The source has voltage amplitude and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced by one with and the source frequency is adjusted to the new resonance value. Then what is the average power delivered by the source?

In an L-R-C series circuit, and (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 550 V. If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

A series circuit consists of an ac source of variable frequency, a \(115 - \Omega)\) resistor, a \(1.25-\mu F\) capacitor, and a 4.50-mH inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency; (b) twice the resonance angular frequency; (c) half the resonance angular frequency. Text Transcription: 115 - Omega 1.25 - muF

In an L-R-C series circuit, L = 0.280 H and C=4.00 ?F. The voltage amplitude of the source is 120 V. (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 1.70 A. What is the resistance R of the resistor? (c) At the resonance angular frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

A Step-Down Transformer. A transformer connected to a 120-V (rms) ac line is to supply 12.0 V (rms) to a portable electronic device. The load resistance in the secondary is \(5.00\ \Omega\). (a) What should the ratio of primary to secondary turns of the transformer be? (b) What rms current must the secondary supply? (c) What average power is delivered to the load? (d) What resistance connected directly across the 120-V line would draw the same power as the transformer? Show that this is equal to \(5.00\ \Omega\) times the square of the ratio of primary to secondary turns.

A Step-Up Transformer. A transformer connected to a 120-V (rms) ac line is to supply 13,000 V (rms) for a neon sign. To reduce shock hazard, a fuse is to be inserted in the primary circuit; the fuse is to blow when the rms current in the secondary circuit exceeds 8.50 mA. (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 mA? (c) What current rating should the fuse in the primary circuit have?

Off to Europe! You plan to take your hair dryer to Europe, where the electrical outlets put out 240 V instead of the 120 V seen in the United States. The dryer puts out 1600 W at 120 V. (a) What could you do to operate your dryer via the 240-V line in Europe? (b) What current will your dryer draw from a European outlet? (c) What resistance will your dryer appear to have when operated at 240 V?

Figure 31.12a shows the crossover network in a loudspeaker system. One branch consists of a capacitor and a resistor in series (the tweeter). This branch is in parallel with a second branch (the woofer) that consists of an inductor and a resistor in series. The same source voltage with angular frequency is applied across each parallel branch. (a) What is the impedance of the tweeter branch? (b) What is the impedance of the woofer branch? (c) Explain why the currents in the two branches are equal when the impedances of the branches are equal. (d) Derive an expression for the frequency that corresponds to the crossover point in Fig. 31.12b

A coil has a resistance of \(48.0\ \Omega\). At a frequency of 80.0 Hz the voltage across the coil leads the current in it by \(52.3^{\circ}\). Determine the inductance of the coil.

tance of the coil. 31.40 .. Five infinite-impedance voltmeters, calibrated to read rms values, are connected as shown in Fig. P31.40. Let and What is the reading of each voltmeter if (a) and (b) 1000 rad>s?

A parallel-plate capacitor having square plates 4.50 cm on each side and 8.00 mm apart is placed in series with an ac source of angular frequency 650 rad s and voltage amplitude 22.5 V, a resistor, and an ideal solenoid that is 9.00 cm long, has a circular cross section 0.500 cm in diameter, and carries 125 coils per centimeter. What is the resonance angular frequency of this circuit? (See Exercise 30.15.)

A toroidal solenoid has 2900 closely wound turns, cross-sectional area \(0.450 \mathrm{~cm}^{2}\), mean radius 9.00 cm, and resistance \(R=2.80 \Omega\). The variation of the magnetic field across the cross section of the solenoid can be neglected. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 V and frequency 365 Hz?

An L-R-C series circuit has \(C=4.80 \mu \mathrm{F}\), L = 0.520 H, and source voltage amplitude V = 56.0 V. The source is operated at the resonance frequency of the circuit. If the voltage across the capacitor has amplitude 80.0 V, what is the value of R for the resistor in the circuit?

A large electromagnetic coil is connected to a 120-Hz ac source. The coil has resistance \(400\ \Omega\) and at this source frequency the coil has inductive reactance \(250\ \Omega\) (a) What is the inductance of the coil? (b) What must the rms voltage of the source be if the coil is to consume an average electrical power of 800 W?

A series circuit has an impedance of and a power factor of 0.720 at 50.0 Hz. The source voltage lags the current. (a) What circuit element, an inductor or a capacitor, should be placed in series with the circuit to raise its power factor? (b) What size element will raise the power factor to unity?

An L-R-C series circuit has \(R=300\ \Omega\). At the frequency of the source, the inductor has reactance \(X_L=900\ \Omega\) and the capacitor has reactance \(X_C=500\ \Omega\). The amplitude of the voltage across the inductor is 450 V. (a) What is the amplitude of the voltage across the resistor? (b) What is the amplitude of the voltage across the capacitor? (c) What is the voltage amplitude of the source? (d) What is the rate at which the source is delivering electrical energy to the circuit?

In an L-R-C series circuit, \(R=300\ \Omega,\ X_C=300\ \Omega\text{, }\text{ and }X_L=500\ \Omega\text{. }\)The average power consumed in the resistor is 60.0 W. (a) What is the power factor of the circuit? (b) What is the rms voltage of the source? Text Transcription: R=300 Omega, X_C = 300 Omega, and X_L = 500 Omega.

A circuit consists of a resistor and a capacitor in series with an ac source that supplies an rms voltage of 240 V. At the frequency of the source the reactance of the capacitor is The rms current in the circuit is 3.00 A. What is the average power supplied by the source?

An L-R-C series circuit consists of a \(50.0-\Omega\) resistor, a \(10.0-\mu \mathrm{F}\) capacitor, a 3.50-mH inductor, and an ac voltage source of voltage amplitude 60.0 V operating at 1250 Hz. (a) Find the current amplitude and the voltage amplitudes across the inductor, the resistor, and the capacitor. Why can the voltage amplitudes add up to more than 60.0 V? (b) If the frequency is now doubled, but nothing else is changed, which of the quantities in part (a) will change? Find the new values for those that do change.

At a frequency \(w_{1}\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to \(w_{2} = 2w_{1}\), what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(w_{3} = w_{1}/3\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance R to form an L-R-C series circuit, what will be the resonance angular frequency of the circuit? Text Transcription: w_1 w_2 = 2w_1 w_3 = w_1/3

A High-Pass Filter. One application of L-R-C series circuits is to high-pass or lowpass filters, which filter out either the low- or high-frequency components of a signal. A highpass filter is shown in Fig. P31.51, where the output voltage is taken across the L-R combination. (The L-R combination represents an inductive coil that also has resistance due to the large length of wire in the coil.) Derive an expression for \(V_{out}/V_{s}\), the ratio of the output and source voltage amplitudes, as a function of the angular frequency w of the source. Show that when w is small, this ratio is proportional to w and thus is small, and show that the ratio approaches unity in the limit of large frequency. Text Transcription: V_out/V_s w

A Low-Pass Filter. Figure P31.52 shows a low-pass filter (see Problem 31.51); the output voltage is taken across the capacitor in an L-R-C series circuit. Derive an expression for \(V_{\text{out }}/V_{\mathrm{s}}\) the ratio of the output and source voltage amplitudes, as a function of the angular frequency \(\omega\) of the source. Show that when \(\omega\) is large, this ratio is proportional to \(\omega^{-2}\) and thus is very small, and show that the ratio approaches unity in the limit of small frequency. Text Transcription: V_out/V_s omega omega^-2

An L-R-C series circuit is connected to an ac source of constant voltage amplitude and variable angular frequency (a) Show that the current amplitude, as a function of is (b) Show that the average power dissipated in the resistor is (c) Show that and are both maximum when , the resonance frequency of the circuit. (d) Graph as a function of for and Compare to the light purple curve in Fig. 31.19. Discuss the behavior of and in the limits

An L-R-C series circuit is connected to an ac source of constant voltage amplitude and variable angular frequency Using the results of Problem 31.53, find an expression for (a) the amplitude of the voltage across the inductor as a function of and (b) the amplitude of the voltage across the capacitor as a function of (c) Graph and as functions of and (d) Discuss the behavior of and in the limits and For what value of is What is the significance of this value of

In an L-R-C series circuit the magnitude of the phase angle is with the source voltage lagging the current. The reactance of the capacitor is and the resistor resistance is 180 . The average power delivered by the source is 140 W. Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

The L-R-C Parallel Circuit. A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency Let the source voltage be given by (a) Show that the instantaneous voltages and at any instant are each equal to and that where is the current through the source and and are the currents through the resistor, the inductor, and the capacitor, respectively. (b) What are the phases of and with respect to Use current phasors to represent and In a phasor diagram, show the phases of these four currents with respect to (c) Use the phasor diagram of part (b) to show that the current amplitude for the current through the source is given by (d) Show that the result of part (c) can be written as

Parallel Resonance. The impedance of an L-R-C parallel circuit was derived in 31.56. (a) Show that at the resonance angular frequency \(\omega_{0}=1 / \sqrt{L C}\), \(I_{C}=I_{L}\), and I is a minimum. (b) Since I is a minimum at resonance, is it correct to say that the power delivered to the resistor is also a minimum at \(\omega=\omega_{0}\)? Explain. (c) At resonance, what is the phase angle of the source current with respect to the source voltage? How does this compare to the phase angle for an L-R-C series circuit at resonance? (d) Draw the circuit diagram for an L-R-C parallel circuit. Arrange the circuit elements in your diagram so that the resistor is closest to the ac source. Justify the following statement: When the angular frequency of the source is \(\omega=\omega_{0}\) there is no current flowing between (i) the part of the circuit that includes the source and the resistor and (ii) the part that includes the inductor and the capacitor, so you could cut the wires connecting these two parts of the circuit without affecting the currents. (e) Is the statement in part (d) still valid if we consider that any real inductor or capacitor also has some resistance of its own? Explain

A resistor and a capacitor are connected in parallel to an ac generator that supplies an rms voltage of 220 V at an angular frequency of Use the results of Problem 31.56. Note that since there is no inductor in the circuit, the term is not present in the expression for Find (a) the current amplitude in the resistor; (b) the current amplitude in the capacitor; (c) the phase angle of the source current with respect to the source voltage; (d) the amplitude of the current through the generator. (e) Does the source current lag or lead the source voltage?

An L-R-C parallel circuit is connected to an ac source of constant voltage amplitude V and variable angular frequency \(\omega\). (a) Using the results of 31.56, find expressions for the amplitudes \(I_R\), \(I_L\), and \(I_C\) of the currents through the resistor, inductor, and capacitor as functions of \(\omega\). (b) Graph \(I_R\), \(I_L\), and \(I_C\) as functions of \(\omega\) for V = 100 V, \(R=200\ \Omega\), L = 2.0 H, and \(C=0.50\ \mu\mathrm{F}\). (c) Discuss the behavior of \(I_L\), and \(I_C\) in the limits \(\omega=0\) and \(\omega=infty\). Explain why \(I_L\), and \(I_C\) behave as they do in these limits. (d) Calculate the resonance frequency (in Hz) of the circuit, and sketch the phasor diagram at the resonance frequency. (e) At the resonance frequency, what is the current amplitude through the source? (f ) At the resonance frequency, what is the current amplitude through the resistor, through the inductor, and through the capacitor?

A \(100-\Omega\) resistor, a \(0.100-\mu \mathrm{F}\) capacitor, and a 0.300-H inductor are connected in parallel to a voltage source with amplitude 240 V. (a) What is the resonance angular frequency? (b) What is the maximum current through the source at the resonance frequency? (c) Find the maximum current in the resistor at resonance. (d) What is the maximum current in the inductor at resonance? (e) What is the maximum current in the branch containing the capacitor at resonance? (f) Find the maximum energy stored in the inductor and in the capacitor at resonance.

You want to double the resonance angular frequency of an L-R-C series circuit by changing only the pertinent circuit elements all by the same factor. (a) Which ones should you change? (b) By what factor should you change them?

An L-R-C series circuit consists of a capacitor, a inductor, and a resistor connected across an ac source of voltage amplitude 15.0 V having variable frequency. (a) Under what circumstances is the average power delivered to the circuit equal to (b) Under the conditions of part (a), what is the average power delivered to each circuit element and what is the maximum current through the capacitor?

In an L-R-C series circuit, the source has a voltage amplitude of 120 V, \(R=80.0 \Omega\) and the reactance of the capacitor is \(480\ \Omega\). The voltage amplitude across the capacitor is 360 V. (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.

An L-R-C series circuit has and (a) For calculate and Using a single set of axes, graph and as functions of time. Include two cycles of on your graph. (b) Repeat part (a) for (c) Repeat part (a) for

CALC The current in a certain circuit varies with time as shown in Fig. P31.65. Find the average current and the rms current in terms o

The Resonance Width. Consider an L-R-C series circuit with a 1.80-H inductor, a capacitor, and a resistor. The source has terminal rms voltage and variable angular frequency (a) What is the resonance angular frequency of the circuit? (b) What is the rms current through the circuit at resonance, (c) For what two values of the angular frequency, and is the rms current half the resonance value? (d) The quantity defines the resonance width. Calculate and the resonance width for and Describe how your results compare to the discussion in Section 31.5.

An inductor, a capacitor, and a resistor are all connected in series across an ac source. If the resistance, inductance, and capacitance are all doubled, by what factor does each of the following quantities change? Indicate whether they increase or decrease: (a) the resonance angular frequency; (b) the inductive reactance; (c) the capacitive reactance. (d) Does the impedance double?

A resistance capacitance and inductance are connected in series to a voltage source with amplitude and variable angular frequency If the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of and

Repeat Problem 31.68 for the case v = v0>2.

Repeat Problem 31.68 for the case \(\omega=2 \omega_{0}\).

A transformer consists of 275 primary windings and 834 secondary windings. If the potential difference across the primary coil is 25.0 V, (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a resistor?

An L-R-C series circuit draws 220 W from a 120-V (rms), 50.0-Hz ac line. The power factor is 0.560, and the source voltage leads the current. (a) What is the net resistance of the circuit? (b) Find the capacitance of the series capacitor that will result in a power factor of unity when it is added to the original circuit. (c) What power will then be drawn from the supply line?

In an L-R-C series circuit the current is given by The voltage amplitudes for the resistor, inductor, and capacitor are and (a) Show that the instantaneous power into the resistor is What does this expression give for the average power into the resistor? (b) Show that the instantaneous power into the inductor is What does this expression give for the average power into the inductor? (c) Show that the instantaneous power into the capacitor is What does this expression give for the average power into the capacitor? (d) The instantaneous power delivered by the source is shown in Section 31.4 to be Show that equals at each instant of time.

CALC (a) At what angular frequency is the voltage amplitude across the resistor in an L-R-C series circuit at maximum value? (b) At what angular frequency is the voltage amplitude across the inductor at maximum value? (c) At what angular frequency is the voltage amplitude across the capacitor at maximum value? (You may want to refer to the results of Problem 31.53.)

Complex Numbers in a Circuit. The voltage across a circuit element in an ac circuit is not necessarily in phase with the current through that circuit element. Therefore the voltage amplitudes across the circuit elements in a branch in an ac circuit do not add algebraically. A method that is commonly employed to simplify the analysis of an ac circuit driven by a sinusoidal source is to represent the impedance as a complex number. The resistance is taken to be the real part of the impedance, and the reactance is taken to be the imaginary part. Thus, for a branch containing a resistor, inductor, and capacitor in series, the complex impedance is where If the voltage amplitude across the branch is we define a complex current amplitude by The actual current amplitude is the absolute value of the complex current amplitude; that is, The phase angle of the current with respect to the source voltage is given by The voltage amplitudes and across the resistance, inductance, and capacitance, respectively, are found by multiplying by and respectively. From the complex representation for the voltage amplitudes, the voltage across a branch is just the algebraic sum of the voltages across each circuit element: The actual value of any current amplitude or voltage amplitude is the absolute value of the corresponding complex quantity. Consider the L-R-C series circuit shown in Fig. P31.75. The values of the circuit elements, the source voltage amplitude, and the source angular frequency are as shown. Use the phasor diagram techniques presented in Section 31.1 to solve for (a) the current amplitude and (b) the phase angle of the current with respect to the source voltage. (Note that this angle is the negative of the phase angle defined in Fig. 31.13.) Now analyze the same circuit using the complex-number approach. (c) Determine the complex impedance of the circuit, Take the absolute value to obtain the actual Z Z, cpx. f impedance of the circuit. (d) Take the voltage amplitude of the source, to be real, and find the complex current amplitude Find the actual current amplitude by taking the absolute value of (e) Find the phase angle of the current with respect to the source voltage by using the real and imaginary parts of as explained above. (f) Find the complex representations of the voltages across the resistance, the inductance, and the capacitance. (g) Adding the answers found in part (f), verify that the sum of these complex numbers is real and equal to 200 V, the voltage of the source.

Problem 7DQ Is it possible for the power factor of an L-R-C series ac circuit to be zero? Justify your answer on physical grounds.

Problem 7E Kitchen Capacitance. The wiring for a refrigerator contains a starter capacitor. A voltage of amplitude 170 V and frequency 60.0 Hz applied across the capacitor is to produce a current amplitude of 0.850 A through the capacitor. What capacitance C is required?

Problem 8DQ In an L-R-C series circuit, can the instantaneous voltage across the capacitor exceed the source voltage at that same instant? Can this be true for the instantaneous voltage across the inductor? Across the resistor? Explain.

Problem 8E (a) Compute the reactance of a 0.450-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50-?F capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450-H inductor equal to that of a 2.50-?F capacitor?

Problem 9DQ In an L-R-C series circuit, what are the phase angle ? and power factor cos ? when the resistance is much smaller than the inductive or capacitive reactance and the circuit is operated far from resonance? Explain.

Problem 9E (a) What is the reactance of a 3.00-H inductor at a frequency of 80.0 Hz? (b) What is the inductance of an inductor whose reactance is 120 ? at 80.0 Hz? (c) What is the reactance of a 4.00-?F capacitor at a frequency of 80.0 Hz? (d) What is the capacitance of a capacitor whose reactance is 120 ? at 80.0 Hz?

Problem 10DQ When an L-R-C series circuit is connected across a 120-V ac line, the voltage rating of the capacitor may be exceeded even if it is rated at 200 or 400 V. How can this be?

Problem 10E A Radio Inductor. You want the current amplitude through a 0.450-mH inductor (part of the circuitry for a radio receiver) to be 1.80 mA when a sinusoidal voltage with amplitude 12.0 V is applied across the inductor. What frequency is required?

In Example 31.6 (Section 31.4), a hair dryer is treated as a pure resistor. But because there are coils in the heating element and in the motor that drives the blower fan, a hair dryer also has inductance. Qualitatively, does including an inductance increase or decrease the values of R, \(I_{r m s}\) and ? Equation Transcription: Text Transcription: I_rms

Problem 11E A 0.180-H inductor is connected in series with a 90.0-? resistor and an ac source. The voltage across the inductor is vL = -(12.0 V)sin[(480 rad/s)t]. (a) Derive an expression for the voltage vR across the resistor. (b) What is vR at t = 2.00 ms?

Problem 12DQ A light bulb and a parallel-plate capacitor with air between the plates are connected in series to an ac source. What happens to the brightness of the bulb when a dielectric is inserted between the plates of the capacitor? Explain.

A \(250-\Omega\) resistor is connected in series with a \(4.80-\mu \mathrm{F}\) capacitor and an ac source. The voltage across the capacitor is \(v_C=(7.60\mathrm{\ V})\sin[(120\mathrm{rad}/\mathrm{s})t]\). (a) Determine the capacitive reactance of the capacitor. (b) Derive an expression for the voltage \(v_R\) across the resistor.

Problem 13DQ A coil of wire wrapped on a hollow tube and a light bulb are connected in series to an ac source. What happens to the brightness of the bulb when an iron rod is inserted in the tube?

Problem 13E A 150-? resistor is connected in series with a 0.250-H inductor and an ac source. The voltage across the resistor is vR = (3.80 V)cos [(720 rad/s)t]. (a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage vL across the inductor.

Problem 14DQ A circuit consists of a light bulb, a capacitor, and an inductor connected in series to an ac source. What happens to the brightness of the bulb when the inductor is omitted? When the inductor is left in the circuit but the capacitor is omitted? Explain.

You have a 200-? resistor, a 0.400-H inductor, and a 6.00-?F capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle ? of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

A circuit consists of a light bulb, a capacitor, and an inductor connected in series to an ac source. Is it possible for both the capacitor and the inductor to be removed and the brightness of the bulb to remain the same? Explain.

Problem 15E The resistor, inductor, capacitor, and voltage source described in Exercise 31.14 are connected to form an L-R-C series circuit. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What is the phase angle of the source voltage with respect to the current? Does the source voltage lag or lead the current? (d) What are the voltage amplitudes across the resistor, inductor, and capacitor? (e) Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source. 31.14 . You have a 200-? resistor, a 0.400-H inductor, and a 6.00-?F capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle ? of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

Problem 25E A series ac circuit contains a 250-? resistor, a 15-mH inductor, a 3.5-?F capacitor, and an ac power source of voltage amplitude 45 V operating at an angular frequency of 360 rad/s. (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

Problem 26E In an L-R-C series circuit the source is operated at its resonant angular frequency. At this frequency, the reactance XC of the capacitor is 200 ? and the voltage amplitude across the capacitor is 600 V. The circuit has R = 300 ?. What is the voltage amplitude of the source?

Problem 27E Analyzing an L - R - C Circuit. You have a 200-? resistor, a 0.400-H inductor, a 5.00-?F capacitor, and a variable-frequency ac source with an amplitude of 3.00 V. You connect all four elements together to form a series circuit. (a) At what frequency will the current in the circuit be greatest? What will be the current amplitude at this frequency? (b) What will be the current amplitude at an angular frequency of 400 rad/s? At this frequency, will the source voltage lead or lag the current?

Problem 28E An L-R-C series circuit is constructed using a 175-? resistor, a 12.5-?F capacitor, and an 8.00-mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 V. (a) At what angular frequency will the impedance be smallest, and what is the impedance at this frequency? (b) At the angular frequency in part (a), what is the maximum current through the inductor? (c) At the angular frequency in part (a), find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value. (d) In part (c), how are the potential differences across the resistor, inductor, and capacitor related to the potential difference across the ac source?

Problem 29E In an L-R-C series circuit, R = 300 ?, L = 0.400 H, and C = 6.00 × 10-8 F. When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 A. (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

Problem 30E An L-R-C series circuit consists of a source with voltage amplitude 120 V and angular frequency 50.0 rad/s, a resistor with R = 400 ?, an inductor with L = 9.00 H, and a capacitor with capacitance C. (a) For what value of C will the current amplitude in the circuit be a maximum? (b) When C has the value calculated in part (a), what is the amplitude of the voltage across the inductor?

Problem 31E In an L-R-C series circuit, R = 150 ?, L = 0.750 H, and C = 0.0180 ?F. The source has voltage amplitude V = 150 V and a frequency equal to the resonance frequency of the circuit. (a) What is the power factor? (b) What is the average power delivered by the source? (c) The capacitor is replaced by one with C = 0.0360 ?F and the source frequency is adjusted to the new resonance value. Then what is the average power delivered by the source?

Problem 32E In an L-R-C series circuit, R = 400 ?, L = 0.350 H, and C = 0.0120 ?F. (a) What is the resonance angular frequency of the circuit? (b) The capacitor can withstand a peak voltage of 670 V. If the voltage source operates at the resonance frequency, what maximum voltage amplitude can it have if the maximum capacitor voltage is not exceeded?

Problem 33E A series circuit consists of an ac source of variable frequency, a 115-? resistor, a 1.25-?F capacitor, and a 4.50-mH inductor. Find the impedance of this circuit when the angular frequency of the ac source is adjusted to (a) the resonance angular frequency; (b) twice the resonance angular frequency; (c) half the resonance angular frequency.

Figure 31.12a shows the crossover network in a loudspeaker system. One branch consists of a capacitor and a resistor in series (the tweeter). This branch is in parallel with a second branch (the woofer) that consists of an inductor and a resistor in series. The same source voltage with angular frequency \(\omega\) is applied across each parallel branch. (a) What is the impedance of the tweeter branch? (b) What is the impedance of the woofer branch? (c) Explain why the currents in the two branches are equal when the impedances of the branches are equal. (d) Derive an expression for the frequency that corresponds to the crossover point in Fig. 31.12b. Equation Transcription: Text Transcription: omega

Problem 37E Off to Europe! You plan to take your hair dryer to Europe, where the electrical outlets put out 240 V instead of the 120 V seen in the United States. The dryer puts out 1600 W at 120 V. (a) What could you do to operate your dryer via the 240-V line in Europe? (b) What current will your dryer draw from a European outlet? (c) What resistance will your dryer appear to have when operated at 240 V?

Problem 39P A coil has a resistance of 48.0 ?. At a frequency of 80.0 Hz the voltage across the coil leads the current in it by 52.3o. Determine the inductance of the coil.

Five infinite-impedance voltmeters, calibrated to read rms values, are connected as shown in Fig. P31.40. Let \(R=200\) \(\Omega\), \(L=0.400\) H, \(C=6.00\) \(\mu \mathrm{F}\), and \(V=30.0\) V. What is the reading of each voltmeter if (a) \(\omega=200\) \(\mathrm{rad} / \mathrm{s}\) and (b) \(\omega=1000\) \(\mathrm{rad} / \mathrm{s}\)? Equation Transcription: Text Transcription: R=200 L=0.400 C=6.00 F V=30.0 omega=200 rad/s omega=1000 rad/s V_1 V_2 V_3 V_4 V_5

Problem 41P CP A parallel-plate capacitor having square plates 4.50 cm on each side and 8.00 mm apart is placed in series with the following: an ac source of angular frequency 650 rad/s and voltage amplitude 22.5 V; a 75.0-? resistor; and an ideal solenoid that is 9.00 cm long, has a circular cross section 0.500 cm in diameter, and carries 125 coils per centimeter. What is the resonance angular frequency of this circuit? (See Exercise 30.15.) 31.15 . The resistor, inductor, capacitor, and voltage source described in Exercise 31.14 are connected to form an L-R-C series circuit. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What is the phase angle of the source voltage with respect to the current? Does the source voltage lag or lead the current? (d) What are the voltage amplitudes across the resistor, inductor, and capacitor? (e) Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source. 31.14 . You have a 200-? resistor, a 0.400-H inductor, and a 6.00-?F capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What is the impedance of the circuit? (b) What is the current amplitude? (c) What are the voltage amplitudes across the resistor and across the inductor? (d) What is the phase angle ? of the source voltage with respect to the current? Does the source voltage lag or lead the current? (e) Construct the phasor diagram.

Problem 42P A toroidal solenoid has 2900 closely wound turns, cross-sectional area 0.450 cm2, mean radius 9.00 cm and resistance R = 2.80 ?. The variation of the magnetic field across the cross section of the solenoid can be neglected. What is the amplitude of the current in the solenoid if it is connected to an ac source that has voltage amplitude 24.0 V and frequency 365 Hz?

Problem 55P In an L-R-C series circuit the magnitude of the phase angle is 54.0o, with the source voltage lagging the current. The reactance of the capacitor is 350 ?, and the resistor resistance is 180 ?. The average power delivered by the source is 140 W. Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

The L-R-C Parallel Circuit. A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency \(\omega\). Let the source voltage be given by \(v=V \cos \omega t\). (a) Show that the instantaneous voltages \(v_{R}\), \(v_{L}\) and \(v_{c}\) at any instant are each equal to and that \(i=i_{R}+i_{L}+i_{C}\), where is the current through the source and \(i_{R}\), \(i_{L}\) and \(i_{C}\) are the currents through the resistor, the inductor, and the capacitor, respectively. (b) What are the phases of \(i_{R}\), \(i_{L}\) and \(i_{C}\) with respect to . Use current phasors to represent \(i_{R}\), \(i_{L}\) and \(i_{C}\). In a phasor diagram, show the phases of these four currents with respect to . (c) Use the phasor diagram of part (b) to show that the current amplitude for the current through the source is given by \(I=\sqrt{I_{R}^{2}+\left(I_{C}-I_{L}\right)^{2}}\). (d) Show that the result of part (c) can be written as \(I=V / Z\), with \(1 / Z=\sqrt{1 / R^{2}+(\omega C-1 / \omega L)^{2}}\). Equation Transcription: Text Transcription: omega v=Vcosomegat v_R v_L v_c i=i_R+i_L+i_C i_R i_L i_C i_R i_L i_C i_R i_L i_C I=sqrt I_R^2+(I_C-I_L)^2 I=V/Z 1/Z=sqrt 1/R^2+(omega C-1/omegaL)^2

Problem 57P Parallel Resonance. The impedance of an L?R?C parallel circuit was derived in Problem. (a) Show that at the resonance angular frequency , and I is a minimum. (b) Since I is a minimum at resonance, is it correct to say that the power delivered to the resistor is also a minimum at ? = ?o? Explain. (c) At resonance, what is the phase angle of the source Current with respect to the source voltage? How does, this compare to the phase angle for an L?R?C series circuit at resonance? (d) Draw the circuit diagram for an L?R?C parallel circuit. Arrange the circuit elements in your diagram so that the resistor is closest to the ac source. Justify the following statement: When the angular frequency of the source is ? = ?o, there is no current flowing between (i) the part of the circuit that includes the source and the resistor and (ii) the part that includes the inductor and the capacitor, so you could cut the wires connecting these two parts of the circuit without affecting the currents. (e) Is the statement in part (d) still valid if we consider that any real inductor or capacitor also has some resistance of its own? Explain. Problem: The L?R?C Parallel Circuit. A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude V and angular frequency ?. Let the source voltage be given by ? = V cos?t. (a) Show that the instantaneous voltages ?R,?L and ?C at any instant are each equal to ? and that i = iR+ iL + iC, where i is the current through the source and iR, iL and iC are the currents through the resistor, the inductor, and the capacitor, respectively. (b) What are the phases of iR, iL and iC with respect to ?? Use current phasors to represent i, iR, iL and iC . In a Phasor diagram, show the phases of these four currents with respect to ?. (c) Use the phasor diagram of part (b) to show that the current amplitude I for the current i through the source is given by (d) Show that the result of part (c) can be written as I = V/Z, with .

A 400-\(\Omega\) resistor and a 6.00-\(\mu \mathrm{F}\) capacitor are connected in parallel to an ac generator that supplies an rms voltage of 220 V at an angular frequency of 360 \(\mathrm{rad} / \mathrm{s}\). Use the results of Problem 31.56. Note that since there is no inductor in the circuit, the \(1 / \omega L\) term is not present in the expression for Z. Find (a) the current amplitude in the resistor; (b) the current amplitude in the capacitor; (c) the phase angle of the source current with respect to the source voltage; (d) the amplitude of the current through the generator. (e) Does the source current lag or lead the source voltage? Equation Transcription: Text Transcription: omega muF rad/s 1/omegaL

An L-R-C parallel circuit is connected to an ac source of constant voltage amplitude and variable angular frequency \(\omega\). (a) Using the results of Problem 31.56, find expressions for the amplitudes \(I_{R}\), \(I_{L}\), and \(I_{C}\) of the currents through the resistor, inductor, and capacitor as functions of \(\omega\). (b) Graph \(I_{R}\), \(I_{L}\), and \(I_{C}\) as functions of \(\omega\) for \(V=100\) V, \(R=200\) \(\Omega\), \(L=2.0\) H,and \(C=0.50\) \(\mu \mathrm{F}\). (c) Discuss the behavior of \(I_{L}\), and \(I_{C}\) in the limits \(\omega=0\) and \(\omega \rightarrow \infty\). Explain why \(I_{L}\), and \(I_{C}\) behave as they do in these limits. (d) Calculate the resonance frequency (in Hz) of the circuit, and sketch the phasor diagram at the resonance frequency. (e) At the resonance frequency, what is the current amplitude through the source? (f ) At the resonance frequency, what is the current amplitude through the resistor, through the inductor, and through the capacitor? Equation Transcription: Text Transcription: omega I_R I_L I_C I_R I_L I_C omega V=100 R=200 omega L=2.0 C=0.50 muF I_L I_C omega=0 omega-infinity

Problem 61P You want to double the resonance angular frequency of an L?R?C series circuit by changing only the pertinent circuit element all by the same factor. (a) Which ones should you change? (b) By what factor should you change them?

An L-R-C series circuit consists of a 2.50-\(\mu \mathrm{F}\) capacitor, a 5.00-mH inductor, and a 75.0-\(\Omega\) resistor connected across an ac source of voltage amplitude 15.0 V having variable frequency. (a) Under what circumstances is the average power delivered to the circuit equal to \(\frac{1}{2} V_{r m s} I_{r m s}\)? (b) Under the conditions of part (a), what is the average power delivered to each circuit element and what is the maximum current through the capacitor? Equation Transcription: Text Transcription: muF Ohms 1 over 2 V_rms I_rms

Problem 63P In an L-R-C series circuit, the source has a voltage amplitude of 120 V, R = 80.0 ?, and the reactance of the capacitor is 480 ?. The voltage amplitude across the capacitor is 360 V. (a) What is the current amplitude in the circuit? (b) What is the impedance? (c) What two values can the reactance of the inductor have? (d) For which of the two values found in part (c) is the angular frequency less than the resonance angular frequency? Explain.

An inductor, a capacitor, and a resistor are all connected in series across an ac source. If the resistance, inductance, and capacitance are all doubled, by what factor does each of the following quantities change? Indicate whether they increase or decrease: (a) the resonance angular frequency; (b) the inductive reactance; (c) the capacitive reactance. (d) Does the impedance double?

Problem 68P A resistance R, capacitance C, and inductance L are connected in series to a voltage source with amplitude V and variable angular frequency ?. If ? = ?0, the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of R, C, L, and V.

Problem 69P Repeat problem for the case ? = ?0/2. Problem: A resistance R, capacitance C, and inductance L are connected in series to a voltage source with amplitude V and variable angular frequency ?. If ? = ?0, the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of R, C, L, and V.

Repeat Problem for the case ? = 2?0 Problem: A resistance R, capacitance C, and inductance L are connected in series to a voltage source with amplitude V and variable angular frequency ?. If ? = ?0, the resonance angular frequency, find (a) the maximum current in the resistor; (b) the maximum voltage across the capacitor; (c) the maximum voltage across the inductor; (d) the maximum energy stored in the capacitor; (e) the maximum energy stored in the inductor. Give your answers in terms of R, C, L, and V.

Problem 71P A transformer consists of 275 primary windings and 834 secondary windings, if the potential difference across the primary coil is 25.0 V, (a) what is the voltage across the secondary coil, and (b) what is the effective load resistance of the secondary coil if it is connected across a 125-? resistor?

Problem 72P An L-R-C series circuit draws 220 W from a 120-V (rms), 50.0-Hz ac line. The power factor is 0.560, and the source voltage leads the current. (a) What is the net resistance R of the circuit? (b) Find the capacitance of the series capacitor that will result in a power factor of unity when it is added to the original circuit. (c) What power will then be drawn from the supply line?

Problem 1DQ Household electric power in most of western Europe is supplied at 240 V, rather than the 120 V that is standard in the United States and Canada. What are the advantages and disadvantages of each system?

Problem 1E You have a special light bulb with a very delicate wire filament. The wire will break if the current in it ever exceeds 1.50 A, even for an instant. What is the largest root-mean-square current you can run through this bulb?

Problem 2DQ The current in an ac power line changes direction 120 times per second, and its average value is zero. Explain how it is possible for power to be transmitted in such a system.

Problem 2E A sinusoidal current i = I cos ?t has an rms value Irms = 2.10 A. (a) What is the current amplitude? (b) The current is passed through a full-wave rectifier circuit. What is the rectified average current? (c) Which is larger: Irms or Irav? Explain, using graphs of i2 and of the rectified current.

Problem 3DQ In an ac circuit, why is the average power for an inductor and a capacitor zero, but not for a resistor?

The voltage across the terminals of an ac power supply varies with time according to Eq. (31.1). The voltage amplitude is What are (a) the root-mean-square potential difference \(V_{\text {rms }}\) and (b) the average potential difference \(V_{a v}\) between the two terminals of the power supply? Equation transcription: Text transcription: V{\text {rms }} V{a v}

Equation (31.14) was derived by using the relationship \(i=d q / d t\) between the current and the charge on the capacitor. In Fig. 31.9a the positive counterclockwise current increases the charge on the capacitor. When the charge on the left plate is positive but decreasing in time, is \(i=d q / d t\) still correct or should it be \(i=-d q / d t ?\) Is \(i=d q / d t\) still correct when the right-hand plate has positive charge that is increasing or decreasing in magnitude? Explain. Equation transcription: Text transcription: i=d q / d t i=-d q / d t ?

Problem 4E A capacitor is connected across an ac source that has voltage amplitude 60.0 V and frequency 80.0 Hz. (a) What is the phase angle ? for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What is the capacitance C of the capacitor if the current amplitude is 5.30 A?

Problem 5DQ Fluorescent lights often use an inductor, called a ballast, to limit the current through the tubes. Why is it better to use an inductor rather than a resistor for this purpose?

Problem 5E An inductor with L = 9.50 mH is connected across an ac source that has voltage amplitude 45.0 V. (a) What is the phase angle ? for the source voltage relative to the current? Does the source voltage lag or lead the current? (b) What value for the frequency of the source results in a current amplitude of 3.90 A?

Equation (31.9) says that \(V_{a b}=L d i / d t\) (see Fig. 31.8a). Using Faraday's law, explain why point is at higher potential than point when is in the direction shown in Fig.31a and is increasing in magnitude. When is counterclockwise and decreasing in magnitude, is \(V_{a b}=L d i / d t\) still correct, or should it \(V_{a b}=-L d i / d t ?\) Is \(V_{a b}=L d i / d t\) still correct when is clockwise and increasing or decreasing in magnitude? Explain. Equation transcription: Text transcription: V{a b}=L d i / d t V{a b}=-L d i / d t ?

Problem 6E A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5.00 mH and C = 3.50 ?F, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

Can a transformer be used with dc? Explain. What happens if a transformer designed for 120-V ac is connected to a 120-V dc line?

Problem 16E A 200-? resistor, 0.900-H inductor, and 6.00-?F capacitor are connected in series across a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What are v, vR, vL, and vC at t = 20.0 ms? Compare vR + vL + vC to v at this instant. (b) What are VR, VL, and VC? Compare V to VR + VL + VC. Explain why these two quantities are not equal.

Problem 17DQ An ideal transformer has N1 windings in the primary and N2 windings in its secondary. If you double only the number of secondary windings, by what factor does (a) the voltage amplitude in the secondary change, and (b) the effective resistance of the secondary circuit change?

Problem 17E In an L-R-C series circuit, the rms voltage across the resistor is 30.0 V, across the capacitor it is 90.0 V, and across the inductor it is 50.0 V. What is the rms voltage of the source?

Problem 18DQ Some electrical appliances operate equally well on ac or dc, and others work only on ac or only on dc. Give examples of each, and explain the differences.

Problem 18E A resistor with R = 300 ? and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 286 W. What is (a) the impedance Z of the circuit; (b) the amplitude of the voltage across the inductor; (c) the power factor?

Problem 19E The power of a certain CD player operating at 120 V rms is 20.0 W. Assuming that the CD player behaves like a pure resistor, find (a) the maximum instantaneous power; (b) the rms current; (c) the resistance of this player.

Problem 20E In an L-R-C series circuit, the components have the following values: L = 20.0 mH, C = 140 nF, and R = 350 ?. The generator has an rms voltage of 120 V and a frequency of 1.25 kHz. Determine (a) the power supplied by the generator and (b) the power dissipated in the resistor.

Problem 21E (a) Show that for an L-R-C series circuit the power factor is equal to R/Z. (b) An L-R-C series circuit has phase angle -31.5o. The voltage amplitude of the source is 90.0 V. What is the voltage amplitude across the resistor?

Problem 22E (a) Use the results of part (a) of Exercise 31.21 to show that the average power delivered by the source in an L-R-C series circuit is given by Pav = Irms2R. (b) An L-R-C series circuit has R = 96.0 ?, and the amplitude of the voltage across the resistor is 36.0 V. What is the average power delivered by the source? 31.21 . (a) Show that for an L-R-C series circuit the power factor is equal to R/Z. (b) An L-R-C series circuit has phase angle -31.5o. The voltage amplitude of the source is 90.0 V. What is the voltage amplitude across the resistor?

An L-R-C series circuit with \(L=0.120\) H, \(R=240\ \Omega\) and \(C=7.30\ \mu\mathrm{F}\) carries an rms current of 0.450 A with a frequency of 400 Hz. (a) What are the phase angle and power factor for this circuit? (b) What is the impedance of the circuit? (c) What is the rms voltage of the source? (d) What average power is delivered by the source? (e) What is the average rate at which electrical energy is converted to thermal energy in the resistor? (f) What is the average rate at which electrical energy is dissipated (converted to other forms) in the capacitor? (g) In the inductor?

Problem 24E An L-R-C series circuit is connected to a 120-Hz ac source that has Vrms = 80.0 V. The circuit has a resistance of 75.0 ? and an impedance at this frequency of 105 ?. What average power is delivered to the circuit by the source?

Problem 34E In an L-R-C series circuit, L = 0.280 H and C = 4.00 ?F. The voltage amplitude of the source is 120 V. (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 1.70 A. What is the resistance R of the resistor? (c) At the resonance angular frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

A Step-Down Transformer. A transformer connected to a 120-V (rms) ac line is to supply 12.0 V (rms) to a portable electronic device. The load resistance in the secondary is 5.00 ?. (a) What should the ratio of primary to secondary turns of the transformer be? (b) What rms current must the secondary supply? (c) What average power is delivered to the load? (d) What resistance connected directly across the 120-V line would draw the same power as the transformer? Show that this is equal to 5.00 ? times the square of the ratio of primary to secondary turns.

Problem 36E A Step-Up Transformer. A transformer connected to a 120-V (rms) ac line is to supply 13,000 V (rms) for a neon sign. To reduce shock hazard, a fuse is to be inserted in the primary circuit; the fuse is to blow when the rms current in the secondary circuit exceeds 8.50 mA. (a) What is the ratio of secondary to primary turns of the transformer? (b) What power must be supplied to the transformer when the rms secondary current is 8.50 mA? (c) What current rating should the fuse in the primary circuit have?

Problem 43P An L-R-C series circuit has C = 4.80 ?F, L = 0.520 H, and source voltage amplitude V = 56.0 V. The source is operated at the resonance frequency of the circuit. If the voltage across the capacitor has amplitude 80.0 V, what is the value of R for the resistor in the circuit?

Problem 44P A large electromagnetic coil is connected to a 120-Hz ac source. The coil has resistance 400 ?, and at this source frequency the coil has inductive reactance 250 ?. (a) What is the inductance of the coil? (b) What must the rms voltage of the source be if the coil is to consume an average electrical power of 800 W?

A series circuit has an impedance of \(60.0 \ \Omega\) and a power factor of 0.720 at 50.0 Hz. The source voltage lags the current. (a) What circuit element, an inductor or a capacitor, should be placed in series with the circuit to raise its power factor? (b) What size element will raise the power factor to unity?

Problem 46P An L?R?C series circuit has R = 300 ?. At the frequency of the source, the inductor has reactance XL = 900 ? and the capacitor has reactance Xc = 500 ?. The amplitude of the voltage across the inductor is 450 V. (a) What is the amplitude of the voltage across the resistor? (b) What is the amplitude of the voltage across the capacitor? (c) What is the voltage amplitude of the source? (d) What is the rate at which the source is delivering electrical energy to the circuit?

Problem 47P In an L-R-C series circuit, R = 300 ?, XC = 300 ?, and XL = 500 ?. The average electrical power consumed in the resistor is 60.0 W. (a) What is the power factor of the circuit? (b) What is the rms voltage of the source?

A circuit consists of a resistor and a capacitor in series with an ac source that supplies an rms voltage of 240 V. At the frequency of the source the reactance of the capacitor is 50.0 ?. The rms current in the circuit is 3.00 A. What is the average power supplied by the source?

Problem 49P An L?R?C series circuit consists of a 50.0-? resistor, a 10.0-µF capacitor, a 3.50-mH inductor, and an ac voltage source of voltage amplitude 60.0 V operating at 1250 Hz. (a) Find the current amplitude and the voltage amplitudes across the inductor, the resistor, and the capacitor. Why can the voltage amplitudes add up to more than 60.0 V? (b) If the frequency is now doubled, but nothing else is changed, which of the quantities in part (a) will change? Find the new values for those that do change.

At a frequency \(?_1\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(?_2=\frac{?_1}{3}\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance R to form an L-R-C series circuit, what will be the resonance angular frequency of the circuit

A High-Pass Filter. One application of series circuits is to high-pass or lowpass filters, which filter out either the low- or high-frequency components of a signal. A high pass filter is shown in Fig. P31.51, where the output voltage is taken across the combination. (The combination represents an inductive coil that also has resistance due to the large length of wire in the coil.) Derive an expression for \(V_{\text {out }} / V_{3}\), the ratio of the output and source voltage amplitudes, as a function of the angular frequency of the source. Show that when is small, this ratio is proportional to and thus is small, and show that the ratio approaches unity in the limit of large frequency. \(V_{o u t}\) Equation transcription: Text transcription: V{\text {out }} / V{3} V{o u t}

A Low-Pass Filter. Figure P31.52 shows a low-pass filter (see Problem 31.51); the output voltage is taken across the capacitor in an series circuit. Derive an expression for \(V_{\text {out }} / V_{3}\), the ratio of the output and source voltage amplitudes, as a function of the angular frequency of the source. Show that when is large, this ratio is proportional to and thus is very small, and show that the ratio approaches unity in the limit of small frequency. \(V_{o u t}\) Equation transcription: Text transcription: V{\text {out }} / V{3} V{o u t}

An series circuit is connected to an ac source of constant voltage amplitude and variable angular frequency (a) Show that the current amplitude, as a function of , is \(i=\frac{V}{\sqrt{R^{2}+(\omega L=1 / \omega C)^{2}}}\) (b) Show that the average power dissipated in the resistor is \(P=\frac{V^{2} R / 2}{R^{2}+(\omega L-1 / \omega C)^{2}}\) (c) Show that and are both maximum when \(\omega=1 / \sqrt{L C}\), the resonance frequency of the circuit. (d) Graph as a function of for \(V=100 \mathrm{~V}, R=200 \Omega, L=2.0 H\), and \(C=0.50 \mu F\). Com- pare to the light purple curve in Fig. 31.19. Discuss the behavior of and in the limits \(\omega=0\) and \(\omega \rightarrow \infty\). Equation transcription: Text transcription: i=\frac{V}{\sqrt{R^{2}+(\omega L=1 / \omega C)^{2}}} P=\frac{V^{2} R / 2}{R^{2}+(\omega L-1 / \omega C)^{2}} \omega=1 / \sqrt{L C} V=100{~V}, R=200 Omega, L=2.0 H C=0.50 \mu F omega=0 omega rightarrow infty ________________

An series circuit is connected to an ac source of constant voltage amplitude and variable angular frequency Using the results of Problem , find an expression for (a) the amplitude \(V_{L}\) of the voltage across the inductor as a function of and (b) the amplitude \(v_{C}\) of the voltage across the capacitor as a function of (c) Graph \(V_{L}\) and \(v_{C}\) as functions of for \(V=100 \mathrm{~V}, R=200 \Omega, L=2.0 H\), and \(C=0.50 \mu F\). (d) Discuss the behavior of \(V_{L}\) and \(v_{C}\) in the limits \(\omega=0\) and \(\omega \rightarrow \infty\). For what value of is \(V_{L}=V_{C} ?\) What is the significance of this value of ? Equation transcription: Text transcription: v{C} V{L} V{L}=V{C} ? V=100{~V}, R=200 Omega, L=2.0 H C=0.50 mu F omega=0 omega rightarrow infty

A 100-? resistor, a 0.100-µF capacitor, and a 0.300-H inductor are connected in parallel to a voltage source with amplitude 240 V. (a) What is the resonance angular frequency? (b) What is the maximum current through the source at the resonance resonance (c) Find the maximum current in the resistor at resonance. (d) What is the maximum current in the inductor at resonance? (e) What is the maximum current in the branch containing the capacitor at resonance? (f) Find the maximum energy stored in the inductor and in the capacitor at resonance.

Problem 64P An L-R-C series circuit has R = 500 ?, L = 2.00 H, C = 0.500 ?F, and V = 100 V. (a) For ? = 800 rad/s, calculate VR, VL, VC, and ?. Using a single set of axes, graph v, vR, vL, and vC as functions of time. Include two cycles of v on your graph. (b) Repeat part (a) for ? = 1000 rad/s. (c) Repeat part (a) for ? = 1250 rad/s.

The current in a certain circuit varies with time as shown in Fig. P31.65. Find the average current and the rms current in terms of \(I_{0}\).

Problem 66P The Resonance Width. Consider an L-R-C series circuit with a 1.80-H inductor, a 0.900-?F capacitor, and a 300-? resistor. The source has terminal rms voltage Vrms = 60.0 V and variable angular frequency ?. (a) What is the resonance angular frequency ?0 of the circuit? (b) What is the rms current through the circuit at resonance, Irms-0? (c) For what two values of the angular frequency, ?1 and ?2, is the rms current half the resonance value? (d) The quantity |?1 - ?2| defines the resonance width. Calculate Irms-0 and the resonance width for R = 300 ?, 30.0 ?, and 3.00 ?. Describe how your results compare to the discussion in Section 31.5.

In an L-R-C series circuit the current is given by \(i=I c o s \omega t\). The voltage amplitudes for the resistor, inductor, and capacitor are \(V_{R}, V_{L}\) and \(V_{C}\). (a) Show that the instantaneous power into the resistor is \(P_{R}=V_{R} I c o s^{2} \omega t=\frac{1}{2} V_{R} I(1+\cos 2 \omega f)\). What does this expression give for the average power into the resistor? (b) Show that the instantaneous power into the inductor is \(P_{L}=-V_{L} I \omega t=\frac{1}{2} V_{L} I_{s i n} 2 \omega t\). What does this expression give for the average power into the inductor? (c) Show that the instantaneous power into the capacitor is \(P_{C}=V_{C} \text { Isinwt } \operatorname{coswt}=\frac{1}{2} V_{C} I s i n 2 \omega t\). What does this expression give for the average power into the capacitor? (d) The instantaneous power delivered by the source is shown in Section 31.4 to be cos(cos ? cos- sin ? sin ). Show that \(P_{R}+P_{L}+P_{C}\) equals at each instant of time. Equation transcription: Text transcription: i=I c o s omega t V{R}, V{L} V{C} P{R}=V{R} I c o s^{2} omega t=frac{1}{2} V_{R} I(1+cos 2 omega f) P{L}=-V{L} I omega t=frac{1}{2} V{L} I{s i n} 2 omega t P{C}=V{C} \text { Isinwt } operatorname{coswt}=frac{1}{2} V{C} I s i n 2 omega t P=V I cos omega t(cos emptyset sin omega t) P{R}+P{L}+P{C}

(a) At what angular frequency is the voltage amplitude across the resistor in an L-R-C series circuit at maximum value? (b) At what angular frequency is the voltage amplitude across the inductor at maximum value? (c) At what angular frequency is the voltage amplitude across the capacitor at maximum value? (You may want to refer to the results of Problem 31.53.)

Complex Numbers in a Circuit. The voltage across a circuit element in an ac circuit is not necessarily in phase with the current through that circuit element. Therefore the voltage amplitudes across the circuit elements in a branch in an ac circuit do not add algebraically. A method that is commonly employed to simplify the analysis of an ac circuit driven by a sinusoidal source is to represent the impedance \(Z\) as a complex number. The resistance \(R\) is taken to be the real part of the impedance, and the reactance \(X=X_{L}-X_{C}\) is taken to be the imaginary part. Thus, for a branch containing a resistor, inductor, and capacitor in series, the complex impedance is \(Z_{\mathrm{cpx}}=R+i X\), where \(i^{2}=-1\). If the voltage amplitude across the branch is \(V_{\mathrm{cpx}}\) we define a complex current amplitude by \(I_{\mathrm{cpx}}=V_{\mathrm{cpx}} / Z_{\mathrm{cpx}}\). The actual current amplitude is the absolute value of the complex current amplitude; that is, \(I=\left(I_{\mathrm{cpx}} * I_{\mathrm{cpx}}\right)^{1 / 2}\). The phase angle \(\phi\) of the current with respect to the source voltage is given by \(\tan \phi=\operatorname{Im}\left(I_{\mathrm{cpx}}\right) / \operatorname{Re}\left(I_{\mathrm{cpx}}\right)\). The voltage amplitudes \(V_{R-\mathrm{cpx}}\), \(V_{L-\mathrm{cpx}}\) and \(V_{C-\mathrm{cpx}}\) across the resistance, inductance, and capacitance, respectively, are found by multiplying \(I_{\mathrm{cpx}}\) by \(R\), \(i X_{L}\), and \(-i X_{C}\) respectively. From the complex representation for the voltage amplitudes, the voltage across a branch is just the algebraic sum of the voltages across each circuit element: \(V_{\mathrm{cpx}}=V_{R-\mathrm{cpx}}+V_{L-\mathrm{cpx}}+V_{C-\mathrm{cpx}}\). The actual value of any current amplitude or voltage amplitude is the absolute value of the corresponding complex quantity. Consider the L-R-C series circuit shown in Fig. P31.75. The values of the circuit elements, the source voltage amplitude, and the source angular frequency are as shown. Use the phasor diagram techniques presented in Section 31.1 to solve for (a) the current amplitude and (b) the phase angle \(\phi\) of the current with respect to the source voltage. (Note that this angle is the negative of the phase angle defined in Fig. 31.13.) Now analyze the same circuit using the complex-number approach. (c) Determine the complex impedance of the circuit, \(Z_{\mathrm{cpx}}\). Take the absolute value to obtain \(Z\) the actual impedance of the circuit. (d) Take the voltage amplitude of the source, \(V_{\mathrm{cpx}}\), to be real, and find the complex current amplitude \(I_{\mathrm{cpx}}\). Find the actual current amplitude by taking the absolute value of \(I_{\mathrm{cpx}}\). (e) Find the phase angle \(\phi\) of the current with respect to the source voltage by using the real and imaginary parts of \(I_{\mathrm{cpx}}\), as explained above. (f) Find the complex representations of the voltages across the resistance, the inductance, and the capacitance. (g) Adding the answers found in part (f), verify that the sum of these complex numbers is real and equal to 200 V, the voltage of the source. Equation Transcription: / ( Text Transcription: Z R X = X_L - X_C Z_cpx = R + iX i^2 = -1 V_cpx I_cpx = V_cpx / Z_cpx I = (I_cpx * I_cpx)^1/2 (tan phi = Im (I_cpx) / Re (I_cpx) V_R-cpx V_L-cpx V_C-cpx I_cpx iX_L -iX_c V_cpx = V_R-cpx + V_L-cpx + V_C-cpx Z_cpx